3.815 \(\int \left (c x^2\right )^{3/2} (a+b x)^2 \, dx\)

Optimal. Leaf size=60 \[ \frac{1}{4} a^2 c x^3 \sqrt{c x^2}+\frac{2}{5} a b c x^4 \sqrt{c x^2}+\frac{1}{6} b^2 c x^5 \sqrt{c x^2} \]

[Out]

(a^2*c*x^3*Sqrt[c*x^2])/4 + (2*a*b*c*x^4*Sqrt[c*x^2])/5 + (b^2*c*x^5*Sqrt[c*x^2]
)/6

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Rubi [A]  time = 0.0381567, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{1}{4} a^2 c x^3 \sqrt{c x^2}+\frac{2}{5} a b c x^4 \sqrt{c x^2}+\frac{1}{6} b^2 c x^5 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^(3/2)*(a + b*x)^2,x]

[Out]

(a^2*c*x^3*Sqrt[c*x^2])/4 + (2*a*b*c*x^4*Sqrt[c*x^2])/5 + (b^2*c*x^5*Sqrt[c*x^2]
)/6

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Rubi in Sympy [A]  time = 11.0313, size = 56, normalized size = 0.93 \[ \frac{a^{2} c x^{3} \sqrt{c x^{2}}}{4} + \frac{2 a b c x^{4} \sqrt{c x^{2}}}{5} + \frac{b^{2} c x^{5} \sqrt{c x^{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(3/2)*(b*x+a)**2,x)

[Out]

a**2*c*x**3*sqrt(c*x**2)/4 + 2*a*b*c*x**4*sqrt(c*x**2)/5 + b**2*c*x**5*sqrt(c*x*
*2)/6

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Mathematica [A]  time = 0.011928, size = 33, normalized size = 0.55 \[ \frac{1}{60} x \left (c x^2\right )^{3/2} \left (15 a^2+24 a b x+10 b^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x^2)^(3/2)*(a + b*x)^2,x]

[Out]

(x*(c*x^2)^(3/2)*(15*a^2 + 24*a*b*x + 10*b^2*x^2))/60

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Maple [A]  time = 0.006, size = 30, normalized size = 0.5 \[{\frac{x \left ( 10\,{b}^{2}{x}^{2}+24\,abx+15\,{a}^{2} \right ) }{60} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)*(b*x+a)^2,x)

[Out]

1/60*x*(10*b^2*x^2+24*a*b*x+15*a^2)*(c*x^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204318, size = 49, normalized size = 0.82 \[ \frac{1}{60} \,{\left (10 \, b^{2} c x^{5} + 24 \, a b c x^{4} + 15 \, a^{2} c x^{3}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^2,x, algorithm="fricas")

[Out]

1/60*(10*b^2*c*x^5 + 24*a*b*c*x^4 + 15*a^2*c*x^3)*sqrt(c*x^2)

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Sympy [A]  time = 2.36483, size = 60, normalized size = 1. \[ \frac{a^{2} c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}}{4} + \frac{2 a b c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} + \frac{b^{2} c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(3/2)*(b*x+a)**2,x)

[Out]

a**2*c**(3/2)*x*(x**2)**(3/2)/4 + 2*a*b*c**(3/2)*x**2*(x**2)**(3/2)/5 + b**2*c**
(3/2)*x**3*(x**2)**(3/2)/6

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GIAC/XCAS [A]  time = 0.207233, size = 47, normalized size = 0.78 \[ \frac{1}{60} \,{\left (10 \, b^{2} x^{6}{\rm sign}\left (x\right ) + 24 \, a b x^{5}{\rm sign}\left (x\right ) + 15 \, a^{2} x^{4}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^2,x, algorithm="giac")

[Out]

1/60*(10*b^2*x^6*sign(x) + 24*a*b*x^5*sign(x) + 15*a^2*x^4*sign(x))*c^(3/2)